Parameter Dependent Robust Control Invariant Sets for LPV Systems with Bounded Parameter Variation Rate

Abstract

Real-time measurements of the scheduling parameter of linear parameter-varying (LPV) systems enables the synthesis of robust control invariant (RCI) sets and parameter dependent controllers inducing invariance. We present a method to synthesize parameter-dependent robust control invariant (PD-RCI) sets for LPV systems with bounded parameter variation, in which invariance is induced using PD-vertex control laws. The PD-RCI sets are parameterized as configuration-constrained polytopes that admit a joint parameterization of their facets and vertices. The proposed sets and associated control laws are computed by solving a single semidefinite programing (SDP) problem. Through numerical examples, we demonstrate that the proposed method outperforms state-of-the-art methods for synthesizing PD-RCI sets, both with respect to conservativeness and computational load.

Figures (4)

• Figure 1: Illustration of PD-RCI sets. (Left: Parameter space) The gray set is $\mathcal{P}$, the current parameter is $p^0$, and the thick black line is $\{p^0\} \oplus \mathcal{R}$, such that $\mathbb{P}(p^0)=\mathrm{CV}\{p^1,p^2\}$. (Right: State space) Red, blue and green sets are $\mathcal{S}(p^0)$, $\mathcal{S}(p^1)$ and $\mathcal{S}(p^2)$ respectively. Hatched region is $\cap_{ p \in \mathbb{P}(p^0)} \mathcal{S}(p)$. Inclusion \ref{['eq:LPV_conditions:3']} implies any $x \in \mathcal{S}(p^0)$ can be driven into the hatched region. The gray set with dot-dashed outline is $\tilde{\mathcal{S}}$ defined in \ref{['eq:tilde_S']}. This set includes $\mathcal{S}(p)$ for all $p \in \mathcal{P}$.
• Figure 2: Results for Example \ref{['sec:Example1']}. (Left: ($x$-$\zeta$ space) The blue set is $\{(x,\zeta) : \zeta \in [-0.25,0.25], x \in \mathcal{S}([0.5+2\zeta,0.5-2\zeta])\}$, and the red set is $X_{\infty} \times \{-0.25\}$, where $X_{\infty}$ is the MRCI set. We obtain larger RCI sets by explicitly accounting for parameter variation. (Right: $x$ space) The gray set is $\mathcal{X}$, and the blue set is $\mathcal{S}(p)$ with $p=[0,1]$ corresponding to $\zeta=-0.25$. The pink set is $\cap_{p^+ \in \mathbb{P}(p)} \mathcal{S}(p^+)$. Initializing $p(0) = [0,1]$ and $x(0) \in \mathcal{S}(p(0))$, we have $x(1) \in \cap_{p^+ \in \mathbb{P}(p)} \mathcal{S}(p^+)$ with $u(0)$ computed as \ref{['eq:cc_control_1']}. The black dotted line is the simulation trajectory, obtained by randomly sampling $p(t)$ while enforcing satisfaction of \ref{['eq:bounded_param_variation']}, and disturbance $w(t)$ sampled randomly from the vertices of $\mathcal{W}$.
• Figure 3: Results for Example \ref{['sec:Example2']}. The green set denotes the RCI set we compute with rows of matrix $C$ representing the normal vectors of a $30$-sided uniform polytope. The blue set is the RCI set obtained using the approach of Gupta2023, and the red set in the RCI set obtained using our approach, with matrix $C$ chosen to be the same as the blue set. Closed-loop trajectories obtained using the vertex feedback law are plotted, illustrating invariance of the green set.
• Figure 4: Projections of the sets $\tilde{\mathcal{S}}$ in grey, and $\mathcal{S}_{\mathrm{PI}}(\bm{W})$ in red. (Top: Projection to $e_y$-$\dot{y}$-$e_{\psi}$ space, Bottom: Projection to $e_y$-$\dot{y}$-$\dot{\psi}$ space.) Blue dots indicate $x(0)$ for several closed-loop trajectories, shown in black, resulting from the parameter-dependent vertex control law. The scheduling parameter sequences satisfy \ref{['eq:bounded_param_variation']} along with $p_2=1/p_1$, and the disturbance sequences are randomly sampled from $\mathcal{W}$.

Theorems & Definitions (11)

• Proposition 1: Strong duality Schneider2013
• Proposition 2
• proof
• Remark 1
• Remark 2
• Remark 3
• Proposition 3
• proof
• Remark 4
• Remark 5